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Convergence in sequences refers to a sequence approaching a specific value as the value of the index, often referred to as \( n \), increases indefinitely. As \( n \) becomes very large, the terms in a convergent sequence become closer and closer to a particular value. This concept is vital in understanding how infinite processes can have finite limits.

A classic example from above is sequence (b): \( s_{n} = \frac{1}{n+1} \). Here, as \( n \rightarrow \infty \), the fraction \( \frac{1}{n+1} \) becomes exceedingly small, ultimately closing in on 0. This is a perfect illustration of a sequence converging to 0 as \( n \rightarrow \infty \). Since the terms are always positive due to the nature of the fraction, this is specifically described as converging to 0 through positive numbers. This highlights how convergence is not just about reaching a number but rather how it approaches it.

• Steadiness towards a single point is key to the concept.
• It can happen with real-valued sequences as seen in sequence \( s_{n} = \frac{1}{n+1} \).
• Look for patterns as \( n \) increases to identify such behavior.

Divergence, particularly to infinity, implies that as \( n \) grows larger, the sequence doesn't settle to a finite limit but grows without bound. Essentially, instead of converging to a specific value, it perpetually escapes towards infinity or negative infinity. This is often observed when the terms increase in magnitude without restriction.

Sequence (a), \( s_{n} = n(n+1) - 1 \), illustrates divergence to infinity. Expanding to \( s_{n} = n^2 + n - 1 \), we see that the \( n^2 \) term clearly dominates as \( n \rightarrow \infty \). The sequence effectively mirrors the growth of \( n^2 \), showcasing divergence to positive infinity (\(+\infty\)).

• Look for terms that perpetually increase as \( n \) grows to identify divergence.
• Dominance of higher degree terms, like \( n^2 \), often hints at divergence to infinity.

Sequence (c) \( s_{n} = 1 - n^2 \) diverges to negative infinity (\(-\infty\)) since the \(-n^2\) term increases in magnitude negatively, showcasing how sequences can have divergent behavior in opposite directions.

Understanding the behavior of a sequence as \( n \rightarrow \infty \) is crucial to determine whether it converges, diverges, or oscillates. This behavior dictates the long-term fate of a sequence, whether it settles towards a number, escapes beyond bounds, or behaves erratically.

Sequence (d), \( s_{n} = \cos \left( \frac{1}{n} \right) \), offers a neat example of behavior as \( n \rightarrow \infty \). As \( n \) increases, \( \frac{1}{n} \) approaches 0, leading the cosine function to converge to \( \cos(0) = 1 \). This illustrates the approach of a sequence towards a finite limit, revealing stable behavior.

• Key factor to watch is the term's change as \( n \) increases.
• Some sequences will converge smoothly, while others, like \( \sin(n)/n \) in sequence (e), fluctuate because of the sine’s inherent oscillatory nature.
• Understanding helps in predicting the limit or potential for infinity.

Sequence (e) shows how the sine function, due to its oscillatory nature, results in a sequence that converges to 0 but oscillates between positive and negative values, explaining how functions intertwine through their trigonometric behavior as \( n \rightarrow \infty \).